I am also having trouble with the save feature. It askes for permission to store data which I give, but when I reopen the game none of my progress is kept.I am playing on Windows 10, with administrative. Using the exe install of the game through the itch launcher.
Hmm, can't say I have tried itch launcher. There is nothing required to install other than Flash Player to run the exe.
The game starts and runs fine, it is the save feature that is not working correctly with the exe.
See the Q&A page on the fan Wiki